This document is dedicated to conducting the confirmatory performance analyses that were proposed for Experiment 3.

Design and Predictions

Design. The design and analysis is a 2 (difficulty: harder than reference vs. easier than reference) X 2 (difference: moderate vs. extreme) X 2 (transition: repeat vs. switch) within-subjects ANOVA on RTs and error rates.

In retrospect, I’m realizing this design doesn’t make sense because, unlike in Experiment 1, decks aren’t associated with a single level of intensity, but rather the risky deck can always have two outcomes. I think trying to coerce this analysis into the difference X difficulty design would introduce a lot of unnecessary complexity.

I’m taking a step back and thinking about what the actual insights are that I want to get out of this performance data. Are RTs and error rates slower and higher for runs of trials with more switches rather than fewer? Is the switch cost greater when switching occurs less frequently? Thinking about it this way compels me to analyze performance based on numbers of switches in a run, regardless of what deck the run comes from. It might also be interesting to explore how the actual decks influence performance, but that would be more of an exploratory performance analysis.

The analysis I’ll conduct to investigate the above is a regression where number of switches in a run and transition type predict RTs (and errors). The prediction is that RTs will be positively associated with number of switches in a run, RTs will be positively predicted by transition such that switch trials have longer RTs (i.e., the switch cost), and there will be an interaction such that the impact of transition on RT will be greater in runs where switching occurs less frequently.

Results

Response Times

Below is the cleaned data:

d <- read.csv('../../../data/dstClean.csv')
d <- d %>% 
  filter(transition != 'startBlock')
N <- d %>% 
  group_by(subject) %>% 
  summarize(n()) %>% 
  nrow(.)
head(d)

The sample size is 236.

Visualize the Results

subjectCellMeans <- d %>% 
  filter(transition != 'startBlock') %>% 
  group_by(subject, outcomeSwitch, transition) %>% 
  summarize(rtime = mean(cuedRt)) 
## `summarise()` has grouped output by 'subject', 'outcomeSwitch'. You can
## override using the `.groups` argument.
subjectCellMeans %>% 
  group_by(outcomeSwitch, transition) %>% 
  summarize(rt = mean(rtime), se = sd(rtime) / sqrt(N)) %>% 
  ggplot(aes(x = factor(outcomeSwitch), y = rt, group = transition)) +
  geom_point(size = 2) +
  geom_line(aes(linetype = transition)) +
  geom_errorbar(aes(ymin = rt - se, ymax = rt + se), width = 0.5) +
  labs(
    x = 'Number of Switches in a Run',
    y = 'Response Time (ms)'
  ) + 
  scale_linetype_discrete(name = 'Transition Type', labels = c('Repeat', 'Switch')) +
  theme_bw() +
  theme(legend.position = c(.9,.5),
        panel.grid = element_blank())
## `summarise()` has grouped output by 'outcomeSwitch'. You can override using the
## `.groups` argument.

Fit a linear model

d$outcomeSwitchC <- d$outcomeSwitch - 8
d$transitionE <- ifelse(d$transition == 'repeat', -0.5, 0.5)
m1 <- lm(cuedRt ~ transitionE * outcomeSwitchC, data = d)
plot(m1)

summary(m1)
## 
## Call:
## lm(formula = cuedRt ~ transitionE * outcomeSwitchC, data = d)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1148.0  -297.5  -134.2   127.3  4732.2 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                1010.1263     1.5153 666.622   <2e-16 ***
## transitionE                 299.2344     3.0306  98.738   <2e-16 ***
## outcomeSwitchC                5.1673     0.4738  10.906   <2e-16 ***
## transitionE:outcomeSwitchC   -8.8521     0.9476  -9.341   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 510.5 on 135264 degrees of freedom
## Multiple R-squared:  0.08695,    Adjusted R-squared:  0.08693 
## F-statistic:  4294 on 3 and 135264 DF,  p-value: < 2.2e-16
confint(m1)
##                                  2.5 %      97.5 %
## (Intercept)                1007.156347 1013.096233
## transitionE                 293.294498  305.174270
## outcomeSwitchC                4.238601    6.095937
## transitionE:outcomeSwitchC  -10.709466   -6.994795
newdata <- data.frame(outcomeSwitch = rep(unique(d$outcomeSwitch), 2), transition = c(rep(-0.5, length(unique(d$outcomeSwitch))), rep(0.5, length(unique(d$outcomeSwitch)))))
newdata$proba <- m1$coefficients[1] + newdata$outcomeSwitch * m1$coefficients[3] + newdata$transition * m1$coefficients[2] + newdata$transition * newdata$outcomeSwitch * m1$coefficients[4]
newdata$transitionE <- newdata$transition
newdata$outcomeSwitchC <- newdata$outcomeSwitch - 8
newdata$transition <- ifelse(newdata$transition == -0.5, 'repeat', 'switch')
newdata <- cbind(newdata, predict(m1, newdata, interval = 'predict'))
ggplot(newdata, aes(x = factor(outcomeSwitch), y = fit, group = transition)) + 
  geom_jitter(data = subjectCellMeans, aes(x = factor(outcomeSwitch), y = rtime, color = factor(transition)), alpha = .2, width = .2, height = 0, show.legend = FALSE) +
  geom_line(size = 2, aes(color = factor(transition))) +
  #geom_ribbon(aes(ymin = lwr, ymax = upr, fill = factor(transition)), alpha = .1, show.legend = FALSE) +
  scale_color_manual(name = 'Transition Type', values = c(`repeat` = 'blue', `switch` = 'red'), labels = c('Repeat', 'Switch')) +
  #scale_fill_manual(values = c(`repeat` = 'blue', `switch` = 'red')) +
  labs(
    x = 'Number of Switches in a Run',
    y = 'Response Time (ms)',
    caption = 'Lines reflect predictions from regression. Points reflect subject-wise cell means.'
  ) +
  theme_bw() +
  theme(legend.position = c(.8,.85),
        panel.grid = element_blank())

Centering at high number of switches per run (12)

d$outcomeSwitchC <- d$outcomeSwitch - 12
d$transitionE <- ifelse(d$transition == 'repeat', -0.5, 0.5)
m1 <- lm(cuedRt ~ transitionE * outcomeSwitchC, data = d)
summary(m1)
## 
## Call:
## lm(formula = cuedRt ~ transitionE * outcomeSwitchC, data = d)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1148.0  -297.5  -134.2   127.3  4732.2 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                1030.7954     2.3838 432.421   <2e-16 ***
## transitionE                 263.8259     4.7676  55.338   <2e-16 ***
## outcomeSwitchC                5.1673     0.4738  10.906   <2e-16 ***
## transitionE:outcomeSwitchC   -8.8521     0.9476  -9.341   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 510.5 on 135264 degrees of freedom
## Multiple R-squared:  0.08695,    Adjusted R-squared:  0.08693 
## F-statistic:  4294 on 3 and 135264 DF,  p-value: < 2.2e-16
confint(m1)
##                                  2.5 %      97.5 %
## (Intercept)                1026.123207 1035.467527
## transitionE                 254.481542  273.170182
## outcomeSwitchC                4.238601    6.095937
## transitionE:outcomeSwitchC  -10.709466   -6.994795

And centering at low number of switches per run (4)

d$outcomeSwitchC <- d$outcomeSwitch - 4
d$transitionE <- ifelse(d$transition == 'repeat', -0.5, 0.5)
m1 <- lm(cuedRt ~ transitionE * outcomeSwitchC, data = d)
summary(m1)
## 
## Call:
## lm(formula = cuedRt ~ transitionE * outcomeSwitchC, data = d)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1148.0  -297.5  -134.2   127.3  4732.2 
## 
## Coefficients:
##                            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                989.4572     2.4686 400.822   <2e-16 ***
## transitionE                334.6429     4.9371  67.781   <2e-16 ***
## outcomeSwitchC               5.1673     0.4738  10.906   <2e-16 ***
## transitionE:outcomeSwitchC  -8.8521     0.9476  -9.341   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 510.5 on 135264 degrees of freedom
## Multiple R-squared:  0.08695,    Adjusted R-squared:  0.08693 
## F-statistic:  4294 on 3 and 135264 DF,  p-value: < 2.2e-16
confint(m1)
##                                 2.5 %     97.5 %
## (Intercept)                984.618866 994.295562
## transitionE                324.966211 344.319602
## outcomeSwitchC               4.238601   6.095937
## transitionE:outcomeSwitchC -10.709466  -6.994795

Looking at the relationship between switches in a run and mean RT (dropping transition)

m2 <- lm(cuedRt ~ outcomeSwitch, data = d)
slope <- round(m2$coefficients[2], 2)

summary(m2)
## 
## Call:
## lm(formula = cuedRt ~ outcomeSwitch, data = d)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1072.4  -329.2  -142.1   153.6  4983.6 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   809.0125     3.8598  209.60   <2e-16 ***
## outcomeSwitch  24.1021     0.4484   53.76   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 528.6 on 135266 degrees of freedom
## Multiple R-squared:  0.02092,    Adjusted R-squared:  0.02091 
## F-statistic:  2890 on 1 and 135266 DF,  p-value: < 2.2e-16
confint(m2)
##                   2.5 %    97.5 %
## (Intercept)   801.44741 816.57759
## outcomeSwitch  23.22335  24.98094
sMeans <- d %>% 
  group_by(subject, outcomeSwitch) %>% 
  summarize(rtime = mean(cuedRt)) 
## `summarise()` has grouped output by 'subject'. You can override using the
## `.groups` argument.
sMeans %>% 
  group_by(outcomeSwitch) %>% 
  summarize(rt = mean(rtime), se = sd(rtime) / sqrt(N)) %>% 
  ggplot(aes(x = factor(outcomeSwitch), y = rt, group = 1)) +
  geom_line() +
  geom_point(size = 2.5) +
  geom_jitter(data = sMeans, aes(x = factor(outcomeSwitch), y = rtime), alpha = .2, width = .1, height = 0) +
  geom_errorbar(aes(ymin = rt - se, ymax = rt + se), width = 0.5) +
  annotate('text', x = 4, y = 1500, label = paste('b = ', slope, sep = '')) +
  labs(
    x = 'Number of Switches in a Run',
    y = 'Response Time (ms)'
  ) +
  theme_bw() +
  theme(panel.grid = element_blank())

Error Rates

Below is the cleaned data:

d <- read.csv('../../../data/dstCleanErrors.csv')

d <- d %>% 
  filter(d$transition != 'startBlock')

N <- d %>% 
  group_by(subject) %>% 
  summarize(n()) %>% 
  nrow(.)
d

The sample size is 236.

Visualize the Results

subjectCellMeans <- d %>% 
  group_by(subject, outcomeSwitch, transition) %>% 
  summarize(error = mean(error)) 
## `summarise()` has grouped output by 'subject', 'outcomeSwitch'. You can
## override using the `.groups` argument.
subjectCellMeans %>% 
  group_by(outcomeSwitch, transition) %>% 
  summarize(err = mean(error), se = sd(error) / sqrt(N)) %>% 
  ggplot(aes(x = factor(outcomeSwitch), y = err, group = transition)) +
  geom_point(size = 2) +
  geom_line(aes(linetype = transition)) +
  geom_errorbar(aes(ymin = err - se, ymax = err + se), width = 0.5) +
  labs(
    x = 'Number of Switches in a Run',
    y = 'Error Rate'
  ) + 
  scale_linetype_discrete(name = 'Transition Type', labels = c('Repeat', 'Switch')) +
  theme_bw() +
  theme(legend.position = c(.2,.8),
        panel.grid = element_blank())
## `summarise()` has grouped output by 'outcomeSwitch'. You can override using the
## `.groups` argument.

Fit a linear model

d$outcomeSwitchC <- d$outcomeSwitch - 8
d$transitionE <- ifelse(d$transition == 'repeat', -0.5, 0.5)
m1 <- lm(error ~ transitionE * outcomeSwitchC, data = d)
plot(m1)

summary(m1)
## 
## Call:
## lm(formula = error ~ transitionE * outcomeSwitchC, data = d)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.04935 -0.04838 -0.03101 -0.02826  0.97358 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                 0.0385388  0.0005403  71.324   <2e-16 ***
## transitionE                 0.0187210  0.0010807  17.324   <2e-16 ***
## outcomeSwitchC              0.0003508  0.0001694   2.071   0.0384 *  
## transitionE:outcomeSwitchC -0.0002167  0.0003389  -0.639   0.5226    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.192 on 150432 degrees of freedom
## Multiple R-squared:  0.002631,   Adjusted R-squared:  0.002612 
## F-statistic: 132.3 on 3 and 150432 DF,  p-value: < 2.2e-16
confint(m1)
##                                    2.5 %       97.5 %
## (Intercept)                 3.747979e-02 0.0395978911
## transitionE                 1.660295e-02 0.0208391426
## outcomeSwitchC              1.874053e-05 0.0006829436
## transitionE:outcomeSwitchC -8.808535e-04 0.0004475527
newdata <- data.frame(outcomeSwitch = rep(unique(d$outcomeSwitch), 2), transition = c(rep(-0.5, length(unique(d$outcomeSwitch))), rep(0.5, length(unique(d$outcomeSwitch)))))
newdata$proba <- m1$coefficients[1] + newdata$outcomeSwitch * m1$coefficients[3] + newdata$transition * m1$coefficients[2] + newdata$transition * newdata$outcomeSwitch * m1$coefficients[4]
newdata$transitionE <- newdata$transition
newdata$outcomeSwitchC <- newdata$outcomeSwitch - 8
newdata$transition <- ifelse(newdata$transition == -0.5, 'repeat', 'switch')
newdata <- cbind(newdata, predict(m1, newdata, interval = 'predict'))
ggplot(newdata, aes(x = factor(outcomeSwitch), y = fit, group = transition)) + 
  geom_jitter(data = subjectCellMeans, aes(x = factor(outcomeSwitch), y = error, color = factor(transition)), alpha = .2, width = .2, height = 0, show.legend = FALSE) +
  geom_line(size = 2, aes(color = factor(transition))) +
  #geom_ribbon(aes(ymin = lwr, ymax = upr, fill = factor(transition)), alpha = .1, show.legend = FALSE) +
  scale_color_manual(name = 'Transition Type', values = c(`repeat` = 'blue', `switch` = 'red'), labels = c('Repeat', 'Switch')) +
  #scale_fill_manual(values = c(`repeat` = 'blue', `switch` = 'red')) +
  labs(
    x = 'Number of Switches in a Run',
    y = 'Error Rate',
    caption = 'Lines reflect predictions from regression. Points reflect subject-wise cell means.'
  ) +
  ylim(0,.2) +
  theme_bw() +
  theme(legend.position = c(.8,.85),
        panel.grid = element_blank()) 
## Warning: Removed 79 rows containing missing values (geom_point).

Looking at the relationship between switches in a run and mean RT (dropping transition)

m2 <- lm(error ~ outcomeSwitch, data = d)
slope <- round(m2$coefficients[2], 5)

summary(m2)
## 
## Call:
## lm(formula = error ~ outcomeSwitch, data = d)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.04754 -0.04144 -0.03839 -0.03381  0.97076 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.0261839  0.0013432  19.493   <2e-16 ***
## outcomeSwitch 0.0015258  0.0001552   9.829   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1922 on 150434 degrees of freedom
## Multiple R-squared:  0.0006418,  Adjusted R-squared:  0.0006351 
## F-statistic: 96.61 on 1 and 150434 DF,  p-value: < 2.2e-16
confint(m2)
##                     2.5 %      97.5 %
## (Intercept)   0.023551134 0.028816608
## outcomeSwitch 0.001221531 0.001830049
sMeans <- d %>% 
  group_by(subject, outcomeSwitch) %>% 
  summarize(error = mean(error)) 
## `summarise()` has grouped output by 'subject'. You can override using the
## `.groups` argument.
sMeans %>% 
  group_by(outcomeSwitch) %>% 
  summarize(err = mean(error), se = sd(error) / sqrt(N)) %>% 
  ggplot(aes(x = factor(outcomeSwitch), y = err, group = 1)) +
  geom_line() +
  geom_point(size = 2.5) +
  geom_jitter(data = sMeans, aes(x = factor(outcomeSwitch), y = error), alpha = .2, width = .1, height = 0) +
  geom_errorbar(aes(ymin = err - se, ymax = err + se), width = 0.5) +
  annotate('text', x = 4, y = .3, label = paste('b = ', slope, sep = '')) +
  labs(
    x = 'Number of Switches in a Run',
    y = 'Error Rate'
  ) +
  theme_bw() +
  theme(panel.grid = element_blank())

 

Analysis Homepage

A work by Dave Braun

dab414@lehigh.edu